{
  "id": "2003.12691",
  "version": "v1",
  "published": "2020-03-28T03:25:34.000Z",
  "updated": "2020-03-28T03:25:34.000Z",
  "title": "On the Ramsey number of a cycle and complete graphs",
  "authors": [
    "Péter Madarasi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we prove that the multicolored Ramsey number $R(G_1,\\dots,G_n,K_{n_1},\\dots,K_{n_r})$ is at least $(\\gamma-1)(\\kappa-1)+1$ for arbitrary connected graphs $G_1,\\dots,G_n$ and $n_1,\\dots,n_r\\in\\mathbb{N}$, where $\\gamma=R(G_1,\\dots,G_n)$ and $\\kappa=R(K_{n_1},\\dots,K_{n_r})$. Erd\\H{o}s at al. conjectured that $R(C_n,K_l)=(n-1)(l-1)+1$ for every $n\\geq l\\geq 3$ except for $n=l=3$. Nikiforov proved this conjecture for $n\\geq 4l+2$. Using the above bound, we derive the following generalization of this result. $R(C_n,K_{n_1},\\dots,K_{n_r})=(n-1)(\\kappa-1)+1$, where $\\kappa=R(K_{n_1},\\dots,K_{n_r})$ and $n\\geq 4\\kappa+2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-28T03:25:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complete graphs",
      "arbitrary connected graphs",
      "multicolored ramsey number",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}